Empirical Analysis of Bank Capital and New Regulatory Requirements for Risks in Trading Portfolios

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We examine the impact of new supervisory standards for bank trading portfolios, additional capital requirements for liquidity risk and credit risk (the Incremental Risk Charge), introduced under Basel 2.5. We estimate risk measures under alternative assumptions on portfolio dynamics (constant level of risk vs. constant positions), rating systems (through-the-cycle vs. point-in-time), for different sectors (asset classes and industry groups), alternative credit risk frameworks (al-ternative dependency structures or factor models) and an extension to a Bayesian framework. We find a potentially material increase in capital requirements, above and beyond that concluded in the far-ranging impact studies conducted by the international supervisors utilizing the participation of a large sample of banks. Results indicate that capital charges are in general higher for either point-in-time ratings or constant portfolio dynamics, with this effect accentuated for financial or sovereign as compared to industrial sectors; and that regulatory is larger than economic capital for the latter, but not for the former sectors. A comparison of the single to a multi-factor credit models shows that capital estimates larger in the latter, and for the financial / sovereign by orders of magnitude vs. industrial or the Basel II model, and that there is less sensitivity of results across sectors and rating systems as compared with the single factor model. Furthermore, in a Bayesian experiment we find that the new requirements may introduce added uncertainty into risk measures as compared to existing approaches.

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  • CR = portf val chng due to spr, rtg migr & dflt Idea w-as to new measures to correct for deficiency in 10-day VaR
  • Empirical Analysis of Bank Capital and New Regulatory Requirements for Risks in Trading Portfolios

    1. 1. Empirical Analysis of Bank Capital and New Regulatory Requirements for Risks in Trading Portfolios Michael Jacobs, Ph.D., CFA Senior Manager Deloitte & Touche LLP Audit and Enterprise Risk Services June 2012
    2. 2. The views expressed herein are those of the authors and do notnecessarily represent the views of the Deloitte and Touche LLP or ofthe Board of Governors of the Federal Reserve System
    3. 3. Outline• Background and Motivation• Introduction and Conclusions• Review of the Literature• Theory and Empirical Methodology• Data and Summary Statistics• Empirical Results and Discussion• Directions for Future Research
    4. 4. Background and Motivation• The 2007-2009 financial crisis was the impetus behind a new set of financial regulations known as “Basel 2.5” and “Basel III”• While credit risk played a central role, not widely known that the epicenter of these losses was in institutions’ trading portfolios• Supervisors require sufficient capital to remain a going concern in a downturn, but trading book losses far exceeded minimums• BCBS introduced 2 metrics: incremental risk charge (IRC) for credit risk in trading books & stressed Value-at-Risk (S-VaR) for estimated mark-to-market losses in a downturn period• IRC measures credit risk in a trading portfolio over capital formation period of 1 year termed the capital horizon (CH)• Period in which unable to rebalance a trading portfolio is the liquidity horizon (LH) set by regulators to be at least 3 months• Constant level of risk (CLR) portfolio dynamics: a certain amount of risk in crises of generating sufficient income to cover losses
    5. 5. Background and Motivation (continued)• CLR implies migration (default) risk only within the LH (CH) so rebalancing causes the risk profile realigned within the year• This influences measured annual default rates upon which probability of default (“PD”) estimates & risk measures are based• Through-the-cycle (“TTC”) ratings quantify credit risk over a business cycle: stable measures not a function of temporary fluctuations in the economy but of obligor fundamentals – Due to this dynamic should observe cyclicality of transitions across the cycle – Ratings agencies or some banks estimate these & are generally favored by supervisors in that they result in less cyclicality in capital measures• Point-in-time (“PIT”) ratings quantify the credit risk over the short term & includes the effects of systematic factors – Observe a greater (less) frequency of transition across (within) ratings – Also used banks concerned about an obligor’s ability to make payments – Examples include vendor models for PD (Moody’s KMV “EDFs”)
    6. 6. Background and Motivation (continued) VIX Volatility Index and C&I Charge-off Rates • A spike in the VIX not seen in 2 decades coincided7.00% VIX Volatility Index C&I Charegoff Rates with massive bank6.00% losses • Clearly, the rise in investor risk5.00% aversion affected traded credit adversely4.00% • Reproduced from: Jacobs, Michael, 2012, Stress3.00% testing credit risk portfolios, Working paper2.00% (presented to Risk- Incisive Media Training1.00% Conference: Credit Risk Management, New York, NY,0.00% March 20th, 2012 ) D e a t 3 6 0 8 9 1 3 0 8 9 1 3 2 8 9 1 3 0 2 8 9 1 6 0 3 8 9 1 3 0 4 8 9 1 3 2 4 8 9 1 3 0 5 8 9 1 3 0 6 8 9 1 3 0 7 8 9 1 3 2 7 8 9 1 3 0 8 9 1 3 6 0 8 9 1 3 0 9 1 3 2 0 9 1 3 0 9 1 3 6 0 2 9 1 0 3 9 1 2 3 9 1 3 0 4 9 1 3 6 0 5 9 1 3 0 6 9 1 3 2 6 9 1 3 0 7 9 1 3 6 0 8 9 1 3 0 9 1 3 2 9 1 3 9 0 2 3 6 1 0 2 1 3 0 2 3 1 0 2 9 3 0 2 3 6 4 0 2 1 3 5 0 2 3 1 5 0 2 3 9 6 0 2 3 6 7 0 2 1 3 8 0 2 3 1 8 0 2 3 9 0 2 3 6 1 0 2
    7. 7. Introduction and Conclusions• We measure the impact of new regulations on firms’ regulatory capital & reach conclusions different and more nuanced as compared to regulatory impact or academic studies• We find evidence that IRC capital may be substantially more than expected by regulatory impact or academic investigations• We find this to vary significantly by portfolio dynamics (CLR, CR), rating system (TTC, PIT), risk model (single vs. multi-factor), sector / asset class• We develop a general theoretical credit risk framework subsuming the structural models of Basel II IRB (B2-IRB) and CreditMetrics (CM) – We implement a version of the industry standard multi-factor generalization of CM in a proprietary R software package that is tractable – We extend the basic B2-IRB and CM models in a Bayesian analysis• In the empirical implementation we utilize extensive data-sets from Datastream (bond and CDS indices in various sectors)
    8. 8. Introduction and Conclusions (continued)• Capital charges are found higher for either PIT / CP ratings / dynamics vs. either TTC or CLR, accentuated for financial or sovereign as compared to industrial sectors• Regulatory capital is found to be larger than economic capital for industrial but not for the financial or sovereign sectors• A comparison of the single to a multi-factor CM model shows that capital estimates larger in the latter and that results do not differ as much across sectors and rating systems – However, for the financial / sovereign by orders of magnitude vs. industrial or the Basel II asymptotic single risk factor (“ASRF”)• We perform a Bayesian implementation of the Basel II asymptotic single risk factor ASRF and the basic CM – Treating correlation matrices and default rates as random rather than as fixed parameters – Illustrates that the new framework potentially adds substantial uncertainty to the capital estimates
    9. 9. Review of the Literature• Gordy (2003): proves that marginal EC calculated using a portfolio model of C-VaR depends on the properties of the portfolio vs. portfolio invariance property of Basel 2 IRB ASRF model• Frey and McNeil (2003) analyze the mathematical structure of portfolio credit risk models, modeling of dependence exploring the role of copulas in latent variable frameworks in popular models• Jimenez-Martin et al (2009) conclude concept of VaR is intended to capture outcomes on a typical day & in market panic sets in & justifies an additional S-VaR measure to correct for the shortfalls• Acharya and Schnabl (2009): later was likely the case because the pre-crisis period was characterized by unusually low volatility• Perignon et al (2010a,b) conclude C-VaR is often inaccurate & has little ability to predict future losses, pre-(post-) crisis data banks systematically over-(under-)estimate trading book capital needs• Berg (2010) calibrates the term structure of risk premia before & in financial crisis finding risk premium term structure flat (increasing) before (during) but long-run mean was of the same magnitude
    10. 10. Review of the Literature (continued)• BCBS (2009a): IRC as a requirement on banks that model specific risk to measure and hold capital against default risk that is incremental to any risk captured in banks’ credit VaR models – Implies IRC should be based CLR at 1-year capital horizon to account for both liquidity risk as well as concentration risk – Market risk is measured as in the current 10 day VaR plus an S-VaR with 99.9th percentile to remain consistent with A-IRB – Still specific risk but only when C-VaR model is only for common factors• BCBS (2009b, 2010a): extensive survey banks internationally concludes that new requirements will result in material increase of regulatory capital (IRC & S-VaR of 51.7% & 28.8%, resp.)• BCBS (2010c): focus on the macroeconomic impact finds a positive long-term effect as justification of the current regime – Higher regulatory capital & depressed lending outweighed by a stronger banking system less vulnerable to financial crises
    11. 11. Review of the Literature (continued)• Kashyap, Stein and Hanson (2010) are in line BCBS (2010c) finding a modest increase in 10% bank capital & 25 BPs in lending rates – But also highlight the flip-side: small changes in banking sector borrowing costs may imply large flows of funds to the “shadow banking system”• Demirguc-Kunt et al (2010) study the financial crisis era equity price reactions of banks with respect to capital ratios & find the ones given more emphasis in Basel III are those most sensitive to equity returns• Varatto (2011): quantitative analysis of the impact of the Basel 2.35 for trading portfolios utilizing various U.S. bond indices – Finds that capital requirements could potentially be much more than estimated by the impact studies conducted by the regulators – Ascribes the diminished capital impact reported by the banks an artifice of an assumed risk reduction due to hedging which in crisis may be questionable• Yavin et al (2011) investigate IRC by constructing alternative TPMs for unsecuritized credit products in the trading book – Argue that banks may need to make discretionary choices in methodology
    12. 12. Theory and Empirical Methodology• Assume a single-period (i.e., fixed time horizon T) model of a credit portfolio consisting of m obligors & an integer-valued state variable: Sij ∈ { 0,1,..., n} i ∈ { 0,1,..., m}• Sij measure of the creditworthiness of each obligor i such that better state is associated with an increasing value & default is coded by : Sij = 0 ⇔ yi = 1 Sij > 0 ⇔ yi = 0• Assume that t = 0 all m obligors in some non-default state & focus on binary outcomes of default and non-default. Define random vector of default indicators for the credit portfolio with joint distribution: Y = ( Y1 ,.., Ym ) ~ p ( y ) = P ( Y1 = y1 ,..., Ym = ym ) y ∈ { 0,1} T m• Denote the marginal default probabilities (or “long-run” PD) by: pi = P ( Yi = 1) ∀ i ∈ { 1,..., m}• Count the number of defaulted obligors at time T with the R.V.: m M ( Y ) = ∑ Yi i =1
    13. 13. Theory and Empirical Methodology (continued)• As realized loss if obligor i defaults is LGDi the random exposure is• EADi, we may define the random overall loss as: … m L = ∑ Yi × LGDi × EADi i =1• We may always posit alternative credit risk models with the same multivariate distribution of Y or S and we will term equivalent any set of models with equality in distribution (E.I.D.)amongst these: d d % EID : Y = Y ∧ S = S %• The simplifying assumption of the exchangeability with respect to default, the mathematical formalism that corresponds to notion of homogenous segments commonly implemented in credit risk models: ( ) ( ) d d   S1 ,.., S m ) = SΠ ( 1) ,.., SΠ ( m ) ⇔ ( Y1 ,.., Ym ) = YΠ ( 1) ,.., YΠ ( m ) (• Implies that for any k ϵ {1,..,m-1} all the possible m!/(k!*(m-k)!)) k dimensional marginal distributions of S are identical
    14. 14. Theory and Empirical Methodology (continued)• Now consider latent variables models, in terms of general structure and there relation to copulas, with an m-dimensional random vector• … and a sequence of cut-off points with the following boundaries: X = ( X 1 ,.., X m ) d1i < ⋅⋅⋅ < d n ∀i ∈ { 1,..., m} d 0 = −∞ d n +1 = +∞ i i i• The state vector and cutoff points are related by: Si = j ⇔  d ij < X i ≤ d ij +1 ∀i ∈ { 1,..., m} , ∀j ∈ { 1,..., n}• It follows that this collection (often interpreted as an obligor’s asset and liability values at the horizon) constitutes a latent variable model (LVM) with respect to the underlying state vector S: ( LVM ( X, d S ) : X i , ( d ij ) ) 1≤ j < n 1≤i ≤ m w.r.t. S = ( S1 ,.., Sm ) T• In this framework, the marginal distribution function of X and the marginal default probability of obligor i are given by : Fi ( x ) = P ( X i ≤ x ) pi = Fi ( d1i )
    15. 15. Theory and Empirical Methodology (continued) • We propose a simple criterion for the equivalence of two LVMs in terms of the marginal distributions of the state vector & the copula • … of X, which facilitates the comparison of various industry risk{ ( LVM ( X, d S ) : X i , ( d ij ) models: ) 0≤ j ≤ n 1≤i ≤ m ( w.r.t. S, LVM ( X, d S ) : X i , ( d ji ) ) 0≤ j ≤ n } w.r.t. S 1≤i ≤ m• The first condition is that that the X’s admit the same copula, the second that marginal distributions of the state vectors coincide: P ( X i ≤ d ij ) = P ( X i ≤ d ji ) ∀j ∈ { 1,.., n} , i ∈ { 1,.., m}• If we are in default mode (i.e., binary sate vector) these reduce to the condition that the marginal PDs are identical in both models• This is only a sufficient condition: in general the converse of this does not hold that the equivalence does not imply the same copulae• JP Morgan’s CreditMetrics (“CM”): assume that the state vector is distributed as a multivariate Gaussian random variable & interpreted as a (standardized) asset value change process over horizon T
    16. 16. Theory and Empirical Methodology (continued)• In CM default threshold is set such that obligor i’s PD is equal to the empirical default frequency of in credit quality & segmentation by• …some kind of PD rating system, either internal by the rating agencies• The dependency structure of the random vector X may estimated in a factor model framework, which means that we express = AZ + Bε + μ X components: Z ∈ R p ~ N p ( 0Ω ) p < m , ε ∈ R m ~ N m ( 0, I M ) Σ=E ( X − E [ X ] ) ( X − E [ X ] ) T  = AΩAT + diag ( σ 2 ,.., σ 2 )  • p<m dimensional random vector Z represents systematic factors, a 1 m standardized independent Gaussian random vector ε idiosyncratic factors, constant matrices A and B contain factor loadings on these• In empirical implementation Z is usually interpreted as observable global, country and industry effects, while ε is an unobserved regression residual in the calibration of the factor weight matrix• In the empirical tests we consider versions of the CM model for both univariate (p=1) or “Base Cm” and multivariate cases (p>1)
    17. 17. Data and Summary Statistics• Estimate the IRC for various test portfolios with different credit rating, maturity sectoral characteristics, a sample portfolio of bond indices using the basic single-factor and multi-factor version of CM• Utilize rating transition matrices of two kinds, TTC from Moody’s Investors Services, and PIT from Kamakura Corporation’s PD model for three sectors (industrial, financial and sovereign) and at two accumulation frequencies, quarterly and annually, in order to test the effect of the CLR vs. the CP assumption• Meant to be portfolios representative of various sectors, daily bond indices compiled by Bank of America-Merrill Lynch and sourced from Datastream spanning the period 1/2/97 to 12/19/11• The industrial Portfolio 1 has 4 rating groups (Aa-Aaa, Baa-A, B-Ba and C-CCC), the financial Portfolio 3 rating groups (Aa-Aaa, A and Baa), and the sovereign Portfolio 3 has 4 rating groups (Aaa, Aa, A and Baa), spanning all maturities
    18. 18. Data and Summary Statistics (continued) Table 3: Point-in-Time (Kamakura KRIS 1990-201) vs. Through-the-Cycle (Moodys DRS 1980-2011) Transition Matrices - Annual Capital Horizon and Quarterly Liquidity Horizon (Constant Level of Risk Assumption) Table 3.1: Industrial Sector, Through-the-Cycle Table 3.4: Industrial Sector, Point-in-Time• … Aaa Aa A Baa Ba B Caa-C Default Aaa Aa A Baa Ba B Caa-C Default Aaa 88.96% 9.69% 1.31% 0.04% 0.01% 0.00% 0.00% 0.00% Aaa 30.29% 15.36% 24.51% 18.48% 6.31% 2.23% 1.24% 1.58% Aa 0.66% 89.23% 9.48% 0.48% 0.10% 0.05% 0.00% 0.00% Aa 18.86% 12.49% 26.73% 25.32% 9.33% 3.23% 1.71% 2.32% A 0.04% 1.34% 91.15% 6.53% 0.69% 0.20% 0.04% 0.01% A 6.93% 7.52% 24.18% 32.65% 14.96% 5.83% 3.33% 4.60% Baa 0.00% 0.17% 3.59% 88.42% 6.45% 1.14% 0.10% 0.09% Baa 1.52% 2.82% 14.26% 32.30% 20.97% 10.24% 6.95% 10.94% Ba 0.00% 0.04% 0.32% 4.30% 83.82% 9.82% 0.65% 1.00% Ba 0.32% 0.89% 6.72% 23.30% 21.71% 13.69% 11.48% 21.89% B 0.00% 0.04% 0.12% 0.37% 4.70% 84.52% 5.13% 5.13% B 0.08% 0.27% 2.78% 13.62% 17.44% 13.72% 13.94% 38.14% Caa-C 0.00% 0.00% 0.02% 0.95% 1.30% 8.63% 65.16% 23.94% Caa-C 0.02% 0.07% 0.88% 5.45% 9.07% 8.68% 10.43% 65.40% Table 3.2: Financial Sector, Through-the-Cycle Table 3.5: Financial Sector, Point-in-Time Aaa Aa A Baa Ba B Caa-C Default Aaa Aa A Baa Ba B Caa-C Default Aaa 87.27% 12.23% 0.53% 0.01% 0.00% 0.00% 0.00% 0.00% Aaa 30.32% 15.32% 24.22% 18.27% 6.28% 2.27% 1.33% 2.01% Aa 0.92% 89.88% 8.85% 0.33% 0.02% 0.00% 0.00% 0.00% Aa 18.87% 12.40% 26.26% 24.93% 9.24% 3.25% 1.81% 3.24% A 0.09% 3.38% 89.88% 6.01% 0.49% 0.04% 0.03% 0.05% A 6.76% 7.25% 23.19% 31.61% 14.60% 5.81% 3.51% 7.27% Baa 0.08% 0.78% 8.94% 82.74% 6.19% 0.99% 0.11% 0.21% Baa 1.50% 2.76% 14.00% 32.06% 21.03% 10.58% 7.70% 10.37% Ba 0.00% 0.15% 1.12% 8.07% 81.04% 7.60% 0.87% 1.15% Ba 0.31% 0.87% 6.60% 23.15% 22.05% 14.57% 13.39% 19.05% B 0.00% 0.12% 0.41% 0.93% 9.79% 76.57% 6.83%5.39% B 0.08% 0.28% 2.90% 14.51% 19.39% 16.28% 18.38% 28.18% Caa-C 0.00% 0.01% 0.04% 0.43% 1.05% 12.96% 47.37% 38.14% Caa-C 0.03% 0.08% 1.03% 6.66% 11.85% 12.30% 16.54% 51.52% Table 3.3: Sovereign Sector, Through-the-Cycle Table 3.6: Sovereign Sector, Point-in-Time Aaa Aa A Baa Ba B Caa-C Default Aaa Aa A Baa Ba B Caa-C Default Aaa 88.96% 9.69% 1.31% 0.04% 0.01% 0.00% 0.00% 0.00% Aaa 30.32% 15.32% 24.22% 18.27% 6.28% 2.27% 1.33% 2.01% Aa 0.66% 89.23% 9.48% 0.48% 0.10% 0.08% 0.00% 0.00% Aa 18.87% 12.40% 26.26% 24.93% 9.24% 3.25% 1.81% 3.24% A 0.04% 1.34% 91.15% 6.53% 0.69% 0.20% 0.04% 0.01% A 6.76% 7.25% 23.19% 31.61% 14.60% 5.81% 3.51% 7.27% Baa 0.00% 0.17% 3.59% 88.42% 6.45% 1.14% 0.10% 0.09% Baa 1.50% 2.76% 14.00% 32.06% 21.03% 10.58% 7.70% 10.37% Ba 0.00% 0.04% 0.32% 4.30% 83.82% 9.82% 0.65% 1.00% Ba 0.31% 0.87% 6.60% 23.15% 22.05% 14.57% 13.39% 19.05% B 0.00% 0.04% 0.12% 0.37% 4.70% 84.52% 5.13% 5.13% B 0.08% 0.28% 2.90% 14.51% 19.39% 16.28% 18.38% 28.18% Caa-C 0.00% 0.00% 0.02% 0.95% 1.30% 8.63% 65.16% 23.94% Caa-C 0.03% 0.08% 1.03% 6.66% 11.85% 12.30% 16.54% 51.52%
    19. 19. Data and Summary Statistics (continued) Table 4: Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1/2/97 to 12/19/11 (Source: Datastream ) Correlation • … Coefficient s between Standard of Systematic Sector Rating Mean Median Deviation Variation Minimum Maximum Correlations Loadings on Systematic Factor Factors Aa-Aaa 0.0377% 0.0274% 0.7167% 18.99 -12.3977% 11.6545% 100.00% 36.07% 8.84% 8.26% 32.92% 55.05% DatastreamPortfolio 1 - Industrials IndexDatastream Baa-A 0.0433% 0.0331% 0.5247% 12.11 -11.5403% 7.4375% 100.00% 8.68% 16.46% 66.26% 48.96% Datastream Total Bond Market IndexIndustrials B-Ba 0.0372% 0.0418% 0.5308% 14.27 -6.0864% 10.8899% 100.00% 78.83% 24.00% 51.39%Indices C-Caa 0.0194% 0.0425% 0.4478% 23.12 -4.7283% 8.3753% 100.00% 29.40% 60.87% 50.61%Portfolio 2 - Aa-Aaa Datastream 0.0286% 0.0185% 0.3977% 13.90 -2.9119% 8.0429% 100.00% 46.23% 14.20% - 88.56% 83.12% Financials IndexDatastream A 0.0411% 0.0097% 0.3375% 8.22 -2.5594% 4.5207% 100.00% 24.17% - 92.23% 80.79%FinancialsIndices Baa 0.0380% 0.0000% 0.7499% 19.74 -3.9653% 10.7130% 100.00% - 51.09% 95.84% 75.10% Sovereigns Index Aaa 0.0334% 0.0097% 0.5006% 14.97 -4.0076% 7.9405% 100.00% 38.39% -32.57% -33.72% 68.09% 88.05% DatastreamPortfolio 3 - Aa 0.0164% 0.0147% 0.0964% 5.87 -0.8925% 0.9980% 100.00% -80.28% -68.16% 91.26% 68.36%Datastream A -0.0259% 0.0000% 2.9434% -113.85 -17.2285% 20.3193% 100.00% 76.78% 75.88% 49.29%SovereignIndices Baa -0.0165% -0.0205% 1.2872% -78.19 -8.9365% 7.4074% 100.00% 74.02% 60.84% 60.09% • High variation in returns relative to the mean & no clear pattern in CV • Correlations vary significantly between sectors and patterns with respect to nearness in rating not always intuitive • Factor loadings for 2-factor CM show that better ratings and financial / sovereign vs. industrial more sensitive to common factors
    20. 20. Data and Summary Statistics (continued)• …
    21. 21. Empirical Results and Discussion Table 5: Incremental Risk Charge (IRC) for CreditMetrics vs. Basel II IRB Models Credit VaR - Credit Capital Expected Credit VaR - Credit Capital Mutlifactor - Mutlifactor• … Loss - Credit Base Credit - Base Credit Credit Credit Expected Credit VaR - Credit Capital Metrics Metrics Metrics Metrics Metrics Loss - Basel 2 Basel 2 IRB - Basel 2 IRB Industrial Sector & Constant Risk 2.60% 6.65% 4.05% 16.45% 13.85% 2.60% 11.05% 8.45% Cycle Ratings Through-the- Financial Sector & Constant Risk 0.00% 0.79% 0.79% 70.50% 70.50% 0.01% 0.34% 0.33% Sovereign Sector & Constant Risk 0.00% 0.47% 0.47% 66.49% 66.49% 0.01% 0.15% 0.14% Industrial Sector & Constant Portfolio 2.32% 6.39% 4.07% 16.81% 14.49% 2.33% 9.95% 7.62% Financial Sector & Constant Portfolio 0.08% 15.11% 15.04% 69.31% 69.24% 0.08% 1.47% 1.39% Sovereign Sector & Constant Portfolio 0.04% 12.09% 12.05% 67.58% 67.54% 0.04% 1.76% 1.72% Industrial Sector & Constant Risk 2.63% 7.17% 4.54% 16.30% 13.68% 2.63% 9.29% 6.66% Point-in-Time Financial Sector & Constant Risk 2.17% 31.93% 29.76% 71.53% 69.36% 2.17% 11.34% 9.17% Ratings Sovereign Sector & Constant Risk 1.44% 28.74% 27.29% 66.76% 65.32% 1.45% 12.47% 11.03% Industrial Sector & Constant Portfolio 3.71% 9.76% 6.05% 16.15% 12.44% 0.93% 6.46% 5.53% Financial Sector & Constant Portfolio 2.17% 33.53% 31.36% 71.09% 68.92% 2.17% 10.12% 7.95% Sovereign Sector & Constant Portfolio 1.09% 26.82% 25.74% 66.98% 65.90% 1.09% 12.15% 11.06%• Generally PIT & CP higher IRC vs. PIT & CR• EC-BCM financial / sovereign EC much smaller only for TTC and CR, else higher by an order of magnitude• Universally EC-MCM higher order of magnitude vs. EC-BCM or RC-B2• Less differences across ratings or portfolio dynamics in the EC-MCM than in the EC- BCM: factor loadings outweighs correlation matrix inputs
    22. 22. Empirical Results and Discussion (continued) Multifactor CreditMetrics Simulated Incremental Risk Charge (IRC) Credit Loss Distribution: Through-the-Cycle Rating Migration Matrix 20 CM-cVar999=16.45% B2-cVar999=11.05% EL=2.50% 15 • … Probability 10 5 0 -0.20 -0.15 -0.10 -0.05 0.00 Credit Losses Datastream Industrial Bond Indices as of 4Q11 (Constant Risk - Moodys Quarterly Transitions 1982-2011) CreditMetrics Simulated Incremental Risk Charge (IRC) Credit Loss Distribution: Through-the-Cycle Rating Migration Matrix Multifactor CreditMetrics Simulated Incremental Risk Charge (IRC) Credit Loss Distribution: Through-the-Cycle Rating Migration Matrix 10000 5 CM-cVar999=70.50% B2-cVar999=0.34% 8000 CM-cVar999=0.79% EL=0.00% 4 6000 B2-cVar999=0.34% 3Probability Probability 4000 EL=0.00% 2 2000 1 0 0 -0.010 -0.008 -0.006 -0.004 -0.002 0.000 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Credit Losses Credit Losses Datastream Financial Bond Indices as of 4Q11 (Constant Risk - Moodys Quarterly Transitions 1982-2011) Datastream Financial Bond Indices as of 4Q11 (Constant Risk - Moodys Quarterly Transitions 1982-2011) • This show the different loss distributions comparisons between EC and RC across industrial & financial sectors, for TTC ratings and CR
    23. 23. Empirical Results and Discussion: Bayesian Estimation Prior Distribution of Correlation: Rating Aa & A Prior Distribution of Correlation: Rating Aa & Ba Prior Distribution of Correlation: Rating Aa & Caa Prior Distribution of Default Rate: Rating Aaa Prior Distribution of Default Rate: Rating Aa Prior Distribution of Default Rate: Rating A 10 4e+05 8e+05 8e+05 8 Probability Density Probability Density Probability Density 8 6 2e+05 4e+05 4e+05 6Probability Density Probability Density Probability Density 6 0e+00 0e+00 0e+00 • … 4 4 0e+00 2e-04 4e-04 6e-04 8e-04 1e-03 0e+00 2e-04 4e-04 6e-04 8e-04 1e-03 0.0000 0.0005 0.0010 0.0015 0.0020 4 Default Rate Default Rate Default Rate Through-the-Cycle / Industrial / Quarterly:Moodys 1982-2011(theta=0.0001,rho=0.010) Through-the-Cycle / Industrial / Quarterly:Moodys 1982-2011(theta=0.0001,rho=0.010) Through-the-Cycle / Industrial / Quarterly:Moodys 1982-2011(theta=0.0002,rho=0.014) 2 2 2 Prior Distribution of Default Rate: Rating Baa Prior Distribution of Default Rate: Rating Ba Prior Distribution of Default Rate: Rating B 4e+05 6000 0 0 0 Probability Density Probability Density Probability Density 15000 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 -0.1 0.0 0.1 0.2 0.3 0.4 4000 2e+05 Correlation Correlation Correlation Industrial Bond Return Indices: Datastream 1997-2011(mean=0.361,stdev=0.014) Industrial Bond Return Indices: Datastream 1997-2011(mean=0.088,stdev=0.016) Industrial Bond Return Indices: Datastream 1997-2011(mean=0.032,stdev=0.146) 2000 5000 Prior Distribution of Correlation: Rating A & Ba Prior Distribution of Correlation: Rating A & Caa Prior Distribution of Correlation: Rating Ba & Caa 0e+00 0 0 8 0.0000 0.0005 0.0010 0.0015 0.0020 0.000 0.005 0.010 0.015 0.020 0.00 0.01 0.02 0.03 0.04 8 8 Default Rate Default Rate Default Rate Through-the-Cycle / Industrial / Quarterly:Moodys 1982-2011(theta=0.0002,rho=0.014) Through-the-Cycle / Industrial / Quarterly:Moodys 1982-2011(theta=0.002,rho=0.045) Through-the-Cycle / Industrial / Quarterly:Moodys 1982-2011(theta=0.012,rho=0.109) 6 6 Prior Distribution of Default Rate: Rating Caa 6Probability Density Probability Density Probability Density 4 1500 Probability Density 4 4 1000 2 2 500 2 0 0 0 0 0.00 0.05 0.10 0.15 -0.1 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Default Rate Through-the-Cycle / Industrial / Quarterly:Moodys 1982-2011(theta=0.069,rho=0.254) Correlation Correlation Correlation Industrial Bond Return Indices: Datastream 1997-2011(mean=0.078,stdev=0.016) Industrial Bond Return Indices: Datastream 1997-2011(mean=0.165,stdev=0.015) Industrial Bond Return Indices: Datastream 1997-2011(mean=0.788,stdev=0.006) • In order to implement this we need prior distributions for correlation and default rate parameters – shown here are the ones for industrial sector and through-the cycle ratings • We implement a non-parametric bootstrap of the bond index history to derive prior distributions for the correlation • In the case of default rates, we fit the history time series to Vasicek distributions, and use those as prior
    24. 24. Empirical Results and Discussion: Bayesian Estimation (continued) Table 6: Incremental Risk Charge (IRC) for CreditMetrics vs. Basel II IRB Models- Bayesian Analysis• … CreditMetrics Basel II IRB CreditMetrics CreditMetrics CreditMetrics CreditMetrics Credit VaR - Basel II IRB Basel II IRB Basel II IRB Basel II IRB Credit VaR - CreditMetrics Credit VaR - Credit VaR - Credit VaR - Credit VaR - Numerical Basel II IRB Credit VaR - Credit VaR - Credit VaR - Credit VaR - Numerical Credit VaR - Posterior Posterior 25th 95th Coefficient of Credit VaR - Posterior Posterior 25th 95th Coefficient of Point Estimate Mean Standard Error Percentile Percentile Variation Point Estimate Mean Standard Error Percentile Percentile Variation Industrial Sector & Constant Risk 6.65% 7.56% 10.68% 0.34% 33.32% 4.3645 11.05% 11.07% 6.82% 0.34% 28.18% 2.7982Cycle RatingsThrough-the- Financial Sector & Constant Risk 0.79% 0.80% 1.60% 0.0005% 2.93% 3.6801 0.34% 0.34% 0.71% 0.0005% 2.14% 6.2132 Sovereign Sector & Constant Risk 0.47% 0.47% 0.73% 0.0020% 1.66% 3.5327 0.15% 0.15% 0.37% 1.96E-05 1.04% 7.0416 Industrial Sector & Constant Portfolio 6.39% 7.63% 10.85% 0.33% 33.69% 4.3721 9.95% 9.93% 6.33% 0.33% 25.95% 2.4247 Financial Sector & Constant Portfolio 15.11% 16.11% 20.95% 0.02% 91.13% 5.6569 1.47% 1.47% 1.91% 0.02% 1.47% 4.5507 Sovereign Sector & Constant Portfolio 12.09% 12.90% 18.04% 0.01% 73.65% 5.7077 1.76% 1.76% 2.16% 0.01% 7.69% 4.3491 Industrial Sector & Constant Risk 7.17% 8.08% 11.46% 0.34% 36.81% 4.5120 9.29% 9.31% 6.09% 0.34% 24.63% 2.4703Point-in-Time Financial Sector & Constant Risk 31.93% 35.52% 32.25% 0.57% 100.00% 2.7989 11.34% 11.39% 6.96% 0.57% 28.65% 2.3323 Ratings Sovereign Sector & Constant Risk 28.74% 31.85% 30.81% 0.37% 100.00% 3.1280 12.47% 12.44% 7.38% 0.37% 30.62% 2.2489 Industrial Sector & Constant Portfolio 9.76% 8.04% 11.33% 0.33% 36.50% 4.4969 6.46% 6.45% 4.78% 0.33% 18.88% 2.7833 Financial Sector & Constant Portfolio 33.53% 35.44% 32.18% 0.54% 100.00% 2.8065 10.12% 10.13% 6.45% 0.54% 26.24% 2.4084 Sovereign Sector & Constant Portfolio 26.82% 29.79% 29.92% 0.28% 100.00% 3.3479 12.15% 12.12% 7.25% 0.28% 29.94% 2.2643• Numerical coefficients of variation in the posterior distribution of the IRC are almost all higher in the CM-EC than the B2-RC model• This pattern is accentuated for the financial / sovereign sector & PIT ratings, but otherwise no clear pattern• Industrials have generally higher CV than financial / sovereign in the case of EC- CM, but this is reversed for B2RC• Overall, in both the CM and regulatory model, there is a high degree of uncertainty in the estimates, more so for TTC vs. PIT ratings
    25. 25. Empirical Results and Discussion: Bayesian Estimation (continued) Bayesian Posterior of CreditMetrics Incremental Risk Charge (IRC) Credit Value-at-Risk: 10 15 20 25 30 CM.cVar999.CV.pst=4.36, CM.cVar999.StdDev.pst=11.08%• … CM.cVar999.q025.pst=0.34% Probability CM.cVar999=6.64% CM.cVar999.mu.pst=7.56% CM.cVar999.q0975.pst=33.32% 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 C-VaR Bayesian Posterior of Basel II IRB Incremental Risk Charge (IRC) Credit Value-at-Risk 10 B2.cVar999.CV.pst=2.36, B2.cVar999.StdDev.pst=6.82% B2.cVar999.q025.pst=0.34% 8 B2.cVar999=11.05% Probability B2.Var999.mu.pst=11.08% 6 B2.cVar999.q0975.pst=28.32% 4 2 0 0.0 0.1 0.2 0.3 0.4 0.5 C-VaR Datastream Industrial Bond Indices as of 4Q11 (Constant Risk: Through-the-Cycle Rating Migration Matrix - Moodys Quarterly Transitions 1982-2011)• For industrials and TTC ratings, this shows the much longer tail in the posterior distribution of the C-VaR for EC vs. RC• Note also the very different shapes: EC-CM has a mode at 0 and rapidly drops off, whereas for RC-B2 it has a mode further right
    26. 26. Directions for Future Research• Implementing a multifactor CM for a realistic, large portfolio of equities/bonds held by a bank comparable to reported to regulators• Estimating the models herein for indices of different instruments such as credit default swaps (CDS)• Extension of the Bayesian estimation framework: expert elicitation of priors and to multifactor CM• Extension of CM to different copula models for the vector of systematic factors (e.g., Student-T, Archimedean)• Consideration of challenger / benchmark credit risk models (e.g., actuarial: CreditRisk+, reduced-form: Jarrow-Chava)• Modeling of stressed VaR by various means (historical simulation vs. extreme value theory-EVT)• Development of a theoretical framework that would endogenously give rise to a process for the measure of risk in a trading portfolio

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